487 dissertation

8/17/2019 487 Dissertation

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UNIVERSITÉ PARIS I – PANTHÉON-SORBONNE

MASTER MMMEF – Parcours Finance

Ramzi MAALOUF

PARTICULARITIES of the COMMODITIES MARKET

Résumé

Ce rapport de stage présente une vue globale du marché des matières premières. L’importance

de ce marché n’a pas cessé d’augmenter ces dernières années. En effet, l’année 2008 était une

année très intéressante, durant laquelle le marché a été témoin d’une envolée des prix du pétrole, de l’or, du cuivre, et des produits agricoles.

La première partie du document introduit le marché et met l’accent sur la courbe forward.

Ensuite, elle se concentre sur la « convenience yield », une notion particulière au monde des

commos. La deuxième partie élabore le sujet des options et volatilités « early expiry », qui est

aussi un concept spécifique au monde des commos. La troisième partie décrit une méthode

utilisée pour extraire la corrélation implicite entre deux sous-jacents à partir d’une option basket. La dernière partie souligne l’importance des options binaires dans le marché des

commos et l’impact qu’a le smile des volatilités sur les prix de ces options.

Rapport de stage présenté leLUNDI 15 SEPTEMBRE a 16H45

JuryGuillaume FOUCHERES BICH Philippe Natixis Université Paris 1

Commodity Derivatives

Lieu du Stage Natixis

47, Quai d'Austerlitz

75013 Paris

France


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Particularities of the Commodities Market

2

Table of Contents1 Futures contracts in commodity markets............................................................................ 3

1.1 Different commodity asset classes ............................................................................. 31.2 Popularity of futures contracts in commodity markets .............................................. 3

1.3 Convenience yield ...................................................................................................... 4

1.4 Comparing futures and forwards................................................................................ 5

1.5 Shapes of the forward curve....................................................................................... 6

1.5.1 Contago .............................................................................................................. 71.5.2 Backwardation.................................................................................................... 8

1.5.3 Impact of the shape of forward curves on option prices .................................... 9

2 Early Expiry Options........................................................................................................ 10

2.1 Implied Volatilities................................................................................................... 10

2.2 Early expiry options: problem description ............................................................... 122.3 Methodology ............................................................................................................ 14

2.4 Numerical example................................................................................................... 15

3 Options on Baskets........................................................................................................... 17

3.1 Pricing options on baskets........................................................................................ 17

3.2 Moment matching..................................................................................................... 18

3.3 Determining implied correlation .............................................................................. 204 Digital options .................................................................................................................. 21

4.1 Popularity of digital options ..................................................................................... 21

4.2 Volatility’s impact on the price of a digital option .................................................. 23

4.3 Smile’s impact on the price of a digital option ........................................................ 25

5 References ........................................................................................................................ 27


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1 Futures contracts in commodity markets

1.1 Different commodity asset classes

The different commodity asset classes are the precious metals (gold, silver, platinum and

palladium), the base metals (copper, nickel, and aluminum), the energy commodities (crudeoil, natural gas …), the agricultural products , also called soft commodities, (wheat, soybean,

coffee …), the CO2 emissions permits and credits, and last but not least, electricity.

1.2 Popularity of futures contracts in commodity markets

The main players in commodity markets are:

1- producers and consumers looking at hedging the commodity’s price risk2- arbitragers looking for a riskless and profitable trading strategy in commodity markets3- investors and speculators looking for an exposure to commodity price moves

In contrast to the equity market, the typical underlying for commodity derivative products is

not the spot or physical commodity, but rather the futures contract on the desired commodity.

For example, in equity markets, a typical call option on a Microsoft share has the share itself

as underlying. In commodity markets on the other hand, a typical call option on Brent has the

Brent futures contract as underlying.

Thanks to the arbitragers (the 2nd class of players in commodity markets), the price of thefutures contract converges to the spot price at the futures maturity. Thus, the payoff of a

European derivative product on a spot commodity is equivalent to a similar product on the

commodity’s futures contract. Mathematically,

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4

So, theoretically, both call options are worth the same. This can be generalized to all types of

European payoffs. Thus, given the premiums are identical (almost identical in practice),

people tend to choose the most liquid asset as underlying. Consequently, in equity markets,stocks are preferred. In commodity markets however, futures are preferred for the followingreasons:

1- they avoid all risks associated with spot trading, namely transportation, storage, andcounterparty risk

2- their prices are readily available on exchanges, unlike spot prices which can only beknown by contacting a number of dealers

3- futures and futures options are generally traded on the same exchange, facilitatinghedging and speculation and reducing arbitrage

1.3 Convenience yield

In equity markets, the absence of arbitrage defines a strict relation between forward and spot

prices. Note that I have purposefully used the term forward instead of futures because I do not

want to consider daily margin calls.

This is not true for commodity forward contracts due to the following reasons:

1- the difficulty of short selling some spot commodities2- the illiquidity of some spot commodities3- the transportation costs4- the storage costs5- the benefits obtained by the ownership of the physical commodity

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Since in the commodities markets, the interest rates are not the main source of uncertainty,they are considered to be deterministic. Thus, futures and forwards are used interchangeably.

This is what I will be assuming for the rest of the document.

1.5 Shapes of the forward curve

We can see that in contrast to equity markets where the entire forward curve can be

determined from the spot price, a commodity’s forward curve is directly read from the market.

Different futures contracts for the same commodity are considered as distinct (though

correlated) underlying assets.


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Consequently, in practice, we work backwards from the forward curve to deduce the implied

convenience yield. Some forward curves may be complex enough requiring a non-constant

convenience yield to explain their shape.

The graph shows the price per barrel on the y-axis and the maturity of the futures contract on the x-axis

So, a deterministic function of time will do the trick.

The generalized formula becomes:

),(

..

T t B

eS F

ds y

t T

t

s

T

t

Once the convenience yield is deduced from the available market quotes, it can be used to

determine the prices of futures that are not quoted on the exchanges.

The most common shapes of forward curves in the commodities markets are the next

subsections.

1.5.1 Contago

If we assume for simplicity that the interest rates and convenience yield are constants,

Contago is the case where (r – y) > 0.

Usually, it is the case where the current world inventories for the commodity in question are

relaxed, and the common feeling in the market is the fear of a possible supply crunch in the

future, stimulating the buying of long dated futures. We therefore see the long dated end of

the curve higher than the short dated one.


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Below is an example of contago:


1.5.2 Backwardation

The forward curve is in backwardation when (r-y) < 0.

Usually, this is the case where the current world inventories are tight, and consumers prefer to

buying now what they will be using in the near future, fearing that inventories will keep ondecreasing. We therefore see the short dated end of the curve higher than the long dated one.

Below is an example of backwardation:



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It is worth noting that although it occurs very rarely, a forward curve may change from

backwardation to contago (or vice versa). This was the case for the Brent and WTI crude oilfutures during May and June 2008. Indeed, consumers feared that oil fields and reserves willsoon be depleted because of the massive demand of emerging countries like China and India.

This pushed the forward curves from contago to backwardation.

1.5.3 Impact of the shape of forward curves on option prices

Speculators, consumers and producers usually buy at the money options. For them, the

important reference is the spot commodity price. However, they usually buy options on thefutures contract with maturity closest to that of the option. These are called standard options

(More on standard options later)

B&S formula for a call option on a future gives us:

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1

})2()1(.{),0(

20

00

volatility futuresthebeing and

T d d T

T K

f

d

with

d N K d N f T BC

basket

T

T

Moreover, because of the absence of arbitrage, the sport price is generally closest to the

futures price with the shortest maturity, called the first nearby. This explains the results shownin the table below

Shape of forwardCurve

Future Compared to strike Call premium

Contango ATM meansT Nearby first f f S K 000

So, the option is actually ITM

Expensive

Backwardation ATM meansT Nearby first f f S K 000

So, the option is actually OTM

Cheap

Table 1-1

Obviously, the opposite is true for a put option.

As a conclusion, backwardation is suitable when investors have a bullish view, and contango

is suitable when they have a bearish view, since in these mentioned cases, the premium they

pay is cheap.


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2 Early Expiry Options

2.1 Implied Volatilities

Since the probability distribution of underlying assets is not lognormal as assumed in the B&S

world, a volatility smile (or skew) exists in the market for a given option maturity. This givesthe implied volatility for each strike. The implied volatility is the volatility that the market

uses to price options for given strikes. They are thus obtained by taking the option premium

from the market and inversing the B&S formula numerically.

Below is an example of a volatility smile:

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

35.0%

4 0

4 5 .

5 1

5 6 .

6 2

6 7 .

7 3

7 8 .

8 4

8 9 .

9 5

1 0 1

1 0 6

1 1 2

1 1 7

1 2 3

1 2 8

1 3 4

1 3 9

1 4 5

1 5 0

1 5 6

The graph shows the volatility smile of WTI crude oil options with 2 years to maturity. On the x-axis, we

have the strikes. On the y-axis, we have the implied volatilities.

If we consider several maturities, we obtain what is called a volatility surface. In order to have

consistent smiles, we should normalize the strikes by dividing them by the futures price. Weobtain what is called a moneyness ratio.

An example of a volatility surface is shown below:


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45%60%

75%90%

105%120%

135%

97

127

160

188

219

251

279309

337370

553735

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

Implied Vol

Moneyness

Maturity

The graph shows the volatility surface of Brent Crude Oil. The maturity is in days.

The volatility matrix corresponding to the above surface is shown below:

MoneynessMaturity in

days45% 60% 75% 90% 105% 120% 135%

97 49.59% 47.52% 44.15% 39.31% 34.92% 33.35% 34.18%127 48.10% 45.63% 43.69% 39.07% 34.79% 33.17% 33.60%160 47.69% 45.00% 42.40% 38.03% 34.31% 32.68% 32.76%188 46.15% 43.83% 41.53% 37.35% 33.84% 32.26% 32.19%219 44.93% 42.90% 40.68% 36.67% 33.35% 31.81% 31.64%251 44.02% 42.12% 39.99% 36.16% 32.95% 31.35% 31.05%

279 41.75% 40.12% 38.04% 34.92% 32.48% 31.38% 31.17%309 41.91% 39.92% 37.58% 34.47% 32.07% 30.96% 30.69%337 39.04% 37.85% 36.18% 33.72% 31.81% 30.85% 30.56%370 38.84% 37.50% 35.84% 33.47% 31.59% 30.66% 30.35%553 39.17% 38.07% 35.94% 32.80% 30.32% 28.86% 28.25%735 37.49% 36.80% 34.53% 31.44% 29.05% 27.56% 26.89%

Table 2-1


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2.2 Early expiry options: problem description

As mentioned in section 1.4.3, market participants tend to buy or sell standard options. Astandard option is an option whose underlying is the futures contract which matures slightly

after the expiry of the option. To clarify matters, we denote:

futureTn the maturity of the nth futures contract listed on the exchange

optionTn the maturity of the standard option on the nth futures contract described above

Note that the standard options are listed on the same exchange as their corresponding futurescontract, and they obviously mature a few days earlier than the futures contract maturity

(option

Tn <futureTn ). The reason for that is to avoid physical delivery of the commodity if

the option is exercised at maturity.

Since for the nth future, standard options with several strikes are listed on the exchange and

are generally liquid products, a volatility smile can be implied for that future. We call this

smile the standard volatility smile of future “n”.

So, for every futures contract (distinguished from other contracts by its maturity), a standard

volatility smile can be implied from the market. The result is a standard volatility matrix for

the commodity in question.

Standard volatility matrix

Moneyness

FutureFuturesMaturity

Options maturity(in days)

80% 90% ATM 110% 120%

Future 1 futureT1 option

1T = 6 49.12% 43.80% 37.84% 34.71% 36.32%


2T = 35 45.59% 41.38% 37.56% 35.28% 35.09%


3T = 66 44.47% 40.62% 37.09% 34.93% 34.29%


4T = 97 42.86% 39.31% 36.10% 34.09% 33.35%

Future 5 futureT5

option

5T = 127 42.45% 39.07% 35.97% 33.98% 33.17%


6T = 160 41.09% 38.03% 35.34% 33.56% 32.68%


7T = 188 40.23% 37.35% 34.83% 33.14% 32.26%

Table 2-2

Sometimes, however, participants may be interested in buying or selling options on futures

contract with longer dated maturities. To be more specific, they are interested in options


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expiring much earlier than their underlying futures contract, hence the name early expiry

options.

As an example, the WTI crude oil options with maturities beyond 1 year are only listed for thematurity months of June and December. So, if the current month is July 2008, and a consumer

wants to hedge against an increase in price of oil in September 2009, he/she will not find the

appropriate call option in the exchange. He/she must buy an Over the Counter early maturity

option. In order to price that option, we need to know the early expiry implied volatility.

So, the problem at hand is the following:

Given a standard volatility matrix (in moneyness format) of future contracts on a certain

commodity, we need to generate an array of early volatility matrices (one early volatility

matrix per future).

Now the question is:

Given the standard volatility matrix shown in the table above, we need to find the volatility of

the option with maturityoption

Tk

on the future “n” as underlying, for all k smaller than n. In

other words, the problem is the one of generating, for each future, the following early

volatility matrix:

Early volatility matrix of future “5”

MoneynessFuture

FuturesMaturity


80% 90% ATM 110% 120%

option

1T = 6 ? ? ? ? ? option

2T = 35 ? ? ? ? ? option

3T = 66 ? ? ? ? ?

option

4T = 97 ? ? ? ? ?

Future 5 futureT5

option

5T = 127 42.45% 39.07% 35.97% 33.98% 33.17%

Table 2-3


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2.3 Methodology

To tackle this problem, we take two main assumptions:

Assumption 1: For a given commodity, ATM instantaneous volatility is stationary

By stationary, we mean that the instantaneous volatility does not depend on time but rather on

the distance to maturity.

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In words, this means that seen from today, the volatility during the last month of the life of anoption is the same as the volatility of an option that expires in one month. Obviously, thisinterpretation applies to all durations other than one month.

Visually,


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.

Assumption 2: For a given futures contract, the standard volatility smile is replicated in itsearly volatility smiles. The reason for that is that traders believe it is the best estimate for the

early volatility smiles. They assume market participants will keep the same skewed

probability distribution for the underlying, independent of the option’s maturity.

Therefore, the methodology to generate an array of early volatility matrices is the following:For future “n”, we first start by computing the ATM early volatilities based on assumption 1.

Then, we compute early ITM and OTM volatilities based on the previously computed earlyATM volatilities and assumption 2.

2.4 Numerical example

The following example better illustrates the algorithm. Suppose our standard volatility matrix

is the one shown in table 2-2. Suppose, moreover, that we want to fill the early volatility

matrix of future “5”, shown in table 2-3.

Note that the last row is exactly the 5th

row of standard volatilities given in the standardvolatility matrix. The remaining volatilities are to be calculated. We show in details the

calculation of the shaded volatilities

We first calculate the 66days early ATM volatility

According to the notation described in the previous section, the first volatility in question is:127

66,0

We know that:

61)(66)(127)( 2127127,662127

66,0

2127

127,0 (1)

T1

T1 T2

T1 T2 T3

Same average ATM volatility

for the specified duration (T1)

Same average ATM volatility

for the specified duration (T2)


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By assumption 1, we have: 127127,66 =61

61,0

Formula (1) thus becomes: 61)(66)(127)( 26161,0212766,02127127,0

Moreover, from the standard volatility matrix, we can directly read the following volatilities:

%83.3466

612)6161,0

(1272)127127,0

(127

66,0,So

37.16%6161,0

:findwe

)3535,0

,(35and)6666,0

,(66couplese between thioninterpolatlinear byThus,

%56.373535,0

%09.376666,0

%97.35127127,0

Figure 1- Illustrating the assumption of stationary ATM volatilities

We now calculate the 66days early 80% volatility (the second volatility in question),

denoted by %80early

By assumption 2, we have: ATM ATM

early

Standard

%80Standard

early

%80

(2)

0 66 127

Early Volatility

(Unknown)

Volatility determined by interpolating standard

volatilities given by the market

(After assuming ATM volatilities are stationary)

Standard volatility given by the market


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The right hand side of the equation can directly be read from the standard volatility matrix.

ATM early , on the other hand, is calculated in the previous step

We therefore have:

%10.41%97.35

45.4283.34%80

early

In the same fashion, the rest of the early volatility matrix is computed. The result is shown inthe table below.

Early volatility matrix of future “5”

Moneyness

FutureFuturesMaturity


80% 90% ATM 110% 120%

option

1T = 6 41.84% 38.51% 35.45% 33.49% 32.70%option

2T = 35 41.54% 38.24% 35.21% 33.26% 32.47%option

3T = 66 41.10% 37.83% 34.83% 32.90% 32.12%

option

4T = 97 41.83% 38.50% 35.45% 33.48% 32.69%

Future 5 futureT5

option

5T = 127 42.45% 39.07% 35.97% 33.98% 33.17%

Table 2-4

3 Options on Baskets

3.1 Pricing options on baskets

Options on a basket of underlying assets are very common in commodity markets. We can

imagine a consumer who uses both natural gas (Nat Gas) and crude oil (WTI) for themanufacturing of his goods. This consumer would be interested in buying a basket call option

to hedge an increase in price rather than buying 2 separate call options, since it is cheaper to

do so. Certainly, this reduction in cost results from a weaker protection, since the basket calloption is only profitable when the average price exceeds the strike.

Mathematically, if we denote

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18

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The two most common ways of pricing basket options are:

1. Monte Carlo2. Moment Matching

I will not address the first method since it is pretty straight forward. The next section will

describe the 2nd

method.

3.2 Moment matching

We know that (under specific conditions), the product of 2 lognormal stochastic process is a

lognormal stochastic process. Their sum however is not. Empirical studies have shown that itcan be approximated as such without loosing a lot of accuracy when calculating the price of

vanilla options. The advantage is the speed of calculation compared to Monte Carlo.

So, if we assume Yt is a lognormal stochastic process, then B&S can be applied to determinethe value of a call on an equally weighted basket.


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19

In fact, B&S gives:

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To do so, we have to pass by an intermediate step:

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T

s s

T

s

e F e E e F

e E F F E

.2,2

0

.2

0

.2,2

0

2

..2

1

0

2,2

0

2,2

0

0

22

0

2

0

2

0

2

0

22

..

.})({


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20

)1(.

.

..2.)(.)({

4

1][

:getweconstant,isthatsimplicityforassumefurtherweIf

....

....

....}.{

2122

21

0

21

0

21

0

2

21

0

21

0

21

0

21

0

21

0

21

2

0

1

0

,2

0

,1

02

2

0

,2

02

1

0

,1

02

02

.,2

0

,1

0

.,2

0

,1

0

.2

1.

2

1.

2

1

,2

0

,1

0

).(

0

.2

1.

2

1

,2

0

,1

0

,2,1

0

T T T

T T

T T

T

Q

dsT T

dsT T

dsdsdsT T

W d Q

dsdsT T T

t

T

t

Q

e

S S

F F e

S

F e

S

F Y E M

e F F e F F

eee F F

e E ee F F F F E

T T

T

s s

T

s

T

s

T

s s s

T

s

T

s

Now that we found M2, we have to find a relation between M2 and basket .

For any exponential martingale, and for Yt in particular, we have:

)2(ln.1

.}{...

2

1

22

.2

1

..2

0

.2

02

.2

1.

0

222

M

M

T

e M e E eY M eY Y

basket

T W QT T W

T basket T basket basket

basket T basket

Now that we found all the parameters we need, B&S formula can readily be applied.

3.3 Determining implied correlation

Sometimes the problem faced by traders is inverted. This means, they observe in the market

(namely brokers, since basket options are not traded on exchanges), prices of products calleddispersions.

A dispersion product between WTI and Nat Gas is the difference between the price of the

separate calls and the price of a call on a basket.In other words:

Dispersion = 0.5 x ( Call(WTI) + Call(Nat Gas) ) – Call(basket)

The trader can easily price the separate calls. So, given the dispersion price, the price of the

call basket is determined.

Once the call basket premium is determined, the B&S formula can be numerically inverted to

find the implied basket volatility, basket .


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21

Then, from equation (2) above, we findT

basket T

basket

eS

F

S

F e M M

T T 222

2

0

,2

0

1

0

,1

02

122

1.

T

S S

F F

eS

F e

S

F M

T T

T T

T T

21

2

0

1

0

,2

0

,1

0

2

2

0

,2

02

1

0

,1

02

2

}

.

.

2

1

].)(.)([4

1

ln{

:usgives(1)equationRearraging(1).equationfrom

ncorrelatioimpliedthedeterminetoneedweeverythinghavewe,MgdetermininAfter

22

21

In most cases, the implied correlation turns out to be different than the historical correlation, just like implied and historical volatilities are different. Implied correlation is interpreted as

the correlation the market is using to price basket options.

From equations (1) and (2), we can see that M2 is an increasing function of , and basket is

an increasing function of M2. We also know that the call value is an increasing function of

volatility. This shows that the price of a call basket is monotonically increasing with .

So, if the market is pricing a high implied correlation, it means there is a common sentiment

that both underlying assets are going to increase together. This creates a high demand on call

baskets for hedging purposes, pushing their price higher.

Once this implied correlation is determined, it can either be used to price other basket options,or it can be used to build a correlation matrix, which is in turn used to price basket options on

more than 2 underlying assets.

4 Digital options

4.1 Popularity of digital options

A European digital (or binary) option with maturity T and strike K on an underlying futures

contract F

T

t, is the option whose payoff at T is K F T T 1 .

Digital or binary options are very common in the commodities market. They are used in the

pricing of popular payoffs, like European barrier options (knock in and knock out options).The reason for that popularity is that the option structure disregards the highly improbable

events with very large payoffs, reducing the premium for the client.

An example of such a payoff is shown below:


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22

We can see that the investor is protected against the rise in the price of the underlying up tothe level of 175%, after which a rebate of 25% of the nominal is paid. Since this event is

considered as highly unlikely, giving away this payoff is not much of a problem for the

investor, and will reduce the overall cost of the structured product.

On the other hand, if the price drops, the client is not exposed to the downside loss unless thedrop is bigger than 25%. In the latter case, the investor experiences a 100% participation loss,

which is capped at 50% of the nominal.

The above payoff is easily structured using vanilla and digital calls and puts.

In fact, if we denote

N Nominal amount invested in the structure

C(K) Vanilla call option with strike K

P(K) Vanilla put option with strike K

DC(K) Digital call option with strike KDP(K) Digital put option with strike K

Payoff = P(50%) – P(75%) – 0.25 x N x DP(75%) + C(100%) – C(175%) – 0.5 x N x

DC(175%)

100755025 175150125

Percentage

Moneyness

Payoff


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23

4.2 Volatility’s impact on the price of a digital option

Unlike vanilla call options, the price of a digital call is not an increasing function of volatility.This can be explained intuitively.

In fact, two cases are to be considered.

Case 1: The digital call is in the money (ITM)

In this case, the intrinsic value of the call is 1, but the time value is negative, since with time,

we risk to fall back below the strike. So, when the option is ITM, we want the volatility to beas low as possible, ideally 0, to remain above the strike. Traders say they are short volatility.

Case 2: The digital call is out of the money (OTM)

In this case, the answer is not very obvious. It depends on how close we are to the strike price.If we are very far below, we are long volatility. If we are very close below, we are short

volatility. Some calculations have to be made to determine the exact level at which digital

call’s Vega changes signs.

Briefly speaking, the digital call option behaves differently than vanilla call options when itcomes to volatility. The reason is that the Digital’s upside potential is limited.

Let’s study the above results mathematically.

)().,0().,0(,

N(0,1)~ z where

.

2

1ln

2

1ln..

).,0(1.),0(

:getwe,QmeasureneutralforwardthetochangingBy

1.

220

2

20

2

0

.2

1

0

00

T

.

00

2

0

d N T Bd z QT B DC So

d T

T K

F

z T F

K W K e F K F

K F QT B E T B DC

e E DC

T

T

T T

W T T T

T

T

T T K F

Q

K F

dt r Q

T

T T

T

T T

T

t

Now, we will study the first derivative of d2 with respect to volatility, which will directly give

us the behavior of the price since N(x) is an increasing function of x.

T T

K

F

d

T

2

1

.

ln

2

0

2


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24

Case 1: The digital call is in the money (ITM)

00ln 20

d K F

T

So, the price of the digital call is always decreasing with volatility. (displays a -ve Vega)

ITM Digital Call Option (Moneyness = 150%)

0

0.2

0.4

0.6

0.8

1

1.2

1 %

7 %

1 3 %

1 9 %

2 5 %

3 1 %

3 7 %

4 3 %

4 9 %

5 5 %

6 1 %

6 7 %

7 3 %

7 9 %

8 5 %

9 1 %

9 7 %

1 0 3 %

1 0 9 %

1 1 5 %

1 2 1 %

1 2 7 %

1 3 3 %

1 3 9 %

1 4 5 %

Volatilities

D i g i t a l C a l l P r i c e

Case 2: The digital call is out of the money (OTM)

otherwised

T

K

F

whend

K

F

T

T

0

ln

200ln

2

0

20

So, the price of the digital call is increasing with volatility. (+ve Vega) up to a certain point,where more volatility will actually hinder the call’s value.


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OTM Digital Call Option (Moneyness = 50%)

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

1 %

9 %

1 7 %

2 5 %

3 3 %

4 1 %

4 9 %

5 7 %

6 5 %

7 3 %

8 1 %

8 9 %

9 7 %

1 0 5 %

1 1 3 %

1 2 1 %

1 2 9 %

1 3 7 %

1 4 5 %

1 5 3 %

1 6 1 %

1 6 9 %

1 7 7 %

1 8 5 %

1 9 3 %

Volatilities

D i g i t a l C a l l P r i c e

4.3 Smile’s impact on the price of a digital option

Traders hedge digital call options by selling the replicating portfolio. The market standard is

to approximately replicate a digital call option with strike K by a vanilla call spread, centered

at K with a very small difference between the 2 strikes K 1 and K 2.

K 1 K 2 K Spot Price

Payoff


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26

In practice, this replicating portfolio rarely has the same value as the formula for the price of adigital call derived in the previous section. This difference is clearly seen when the volatilitysmile is not flat.

Intuitively, when the volatility skew is upward sloping, the replicating portfolio consists of

buying a large number of calls with strike K1 and selling the same number of calls with strike

K2. Since C(K1) is priced with a volatility V1 a bit smaller than V2 ( which is used to priceC(K2) ), the replicating portfolio is worth less than the case where V1 = V2 (when the

volatility smile is flat). The opposite can be said when the volatility is downward sloping.

So, given the same parameters (Maturity, Strike, Volatility, and interest rates), and the

different volatility smiles shown below,

We can say, DC(Smile 1) > DC(Smile 2) > DC(Smile 3)

To show this result mathematically:

smilevolatilityof SlopeCall Vanillaof Vega

K

d N T B

K dK

K d r T K S C

K

r T K S C

K

r T K K S C r T K K S C r T K K S DC

.,,,,,,,,

,,,,

.2

,,,,lim,),(,,

)().,0(

0

2

First, we observe that when the smile is flat (B&S world), the slope is zero, and the equation

reduces to the formula obtained in the previous section.

Second, since the Vega of a vanilla call option is always positive, the price is only affected by

the sign of the slope parameter. In the case it is positive, the price decreases and vice versa.

Smile 1

Smile 2

Smile 3

Strikes

Volatilities


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5 References

1- Commodities and Commodity Derivatives: Modeling and Pricing for Agriculturals, Metals, and Energy by Helyette ,Prof. Geman

2- Options, Futures and Other Derivatives, by John C. Hull3- Commodity trade definitions and pricing, ECDA Analytics, Natixis

4- Interest Rate Models –Theory and Practice, by Damiano Brigo · Fabio Mercurio

487 dissertation

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